3.4 Vibrating Rotor
61
3.4 Vibrating Rotor
A rough approximation is to assume that vibration and rotation do not interact. The
total Hamiltonian is
H T = H V + H R
(3.26)
the total energy
E T = E V + E R
(3.27)
and the total wavefunction
ψ T = ψ V ψ R
(3.28)
Actually, when one compares a theoretical spectrum with the experimental
spectrum of a single isotopologue, two main discrepancies appear:
1. Where one expects a single line, one finds a series of nearly equidistant lines
whose intensity increases with the frequency. The less intense lines may be
attributed to the excited states of vibration (υ
> 0).
2. When one considers the most intense lines of each series which are assumed to
belong to the ground vibrational state (υ = 0), one observes that they are not
equally spaced but that there is a deviation increasing with (J + 1)
3 . It is explained
by the fact that the molecule is not rigid. Going back to classical mechanics, it
is obvious that the centrifugal force increases with the angular velocity and,
therefore, the bond length is expected to increase with J.
As the vibration motion is much faster than the rotation motion, one has to take
into account the mean value of the bond length
r
n
V
=
ψ
∗
V r
n ψ V dτ
(3.29)
H T ψ V ψ R = H V ψ V ψ R +
hP
2
8π 2 μ
1
r 2 ψ V ψ R
= E T ψ V ψ R
(3.30)
Multiplying on the left by ψ V gives
E V ψ R +
hP
2
8π 2 μ
ψ V |
1
r 2 |ψ V
H
eff
R
= E T ψ R
(3.31)